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Mathematical and Optimization Analysis of a Miniature Stirling Cryocooler

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Mathematical and Optimization Analysis of a Miniature Stirling Cryocooler View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by ethesis@nitr MATHEMATICAL AND OPTIMIZATION ANALYSIS OF A MINIATURE STIRLING CRYOCOOLER Thesis submitted in partial fulfillment of the requirements for the degree of Bachelor of Technology (B. Tech) In Mechanical Engineering By Yatin Chhabra (107ME025) Devaraj. V (107ME015) Under the guidance of Prof. R.K. Sahoo NATIONAL INSTITUTE OF TECHNOLOGY ROURKELA Page | i Certificate of Approval This is to certify that the thesis entitled Mathematical and Optimization Analysis of a Miniature Stirling Cryo-cooler submitted by Mr. Yatin Chhabra and Mr. Devaraj. V has been carried out under my supervision in partial fulfillment of the requirements for the Degree of Bachelors of Technology (B. Tech) in Mechanical Engineering at National Institute of Technology Rourkela, and this work has not been submitted elsewhere before for any other academic degree/diploma. .................................................. Dr. R. K. Sahoo Professor Department of Mechanical Engineering National Institute of Technology, Rourkela Page | ii Abstract In the given report, a comprehensive analytical model of the working of a miniature Stirling Cyo-cooler is presented. The motivation of the study is to determine the optimum geometrical parameters of a cryo-cooler such as compressor length, regenerator diameter, expander diameter, and expander stroke. In the first part of the study, an ideal analysis is carried out using the Stirling Cycle and basic thermodynamics equations. Using these equations, rough geometrical parameters are found out. In the second part of the study, a more comprehensive Schmidt’s analysis is carried out. In this analysis, pressure and volume variations are considered sinusoidal and based on these, various equations regarding efficiency and COP are derived. Various graphs are generated in MATLAB plotting Refrigeration and Work done w.r.t to various geometrical parameters. With the help of these graphs, the net refrigeration obtained is calculated for a given geometry of cryo-cooler.. This model provides a more accurate picture of the cryo-cooler. However in this analysis, regenerator efficiency is considered 100 % which is not true in practical cases. In the third and final part of the study, optimization of regenerator is carried out. This part is based on Ackermann’s analysis in which various looses taking place inside a regenerator are considered and accounted for. These looses are minimized using an iterative cycle and optimum regenerator dimensions are obtained. Thus the geometrical results obtained from the third part of the study are expected to be most accurate as it accounts for most of the looses taking place inside a cryo-cooler. Page | iii Acknowledgment I was privileged to have worked under the supervision of Prof. R.K. Sahoo. His invaluable advise, enthusiasm and support throughout my project provided me with the motivation to stay on track and fulfill my potential. I would also like to thank Prof. S.K. Sarangi for his timely advise and encouragement which helped me get through the difficult periods during the project. In the end I would also like to extend my gratitude to the current Masters Students of Mechanical Engineering department for their help regarding the use of computational software’s like FLUENT and MATLAB. Page | iv Table of Contents 1) INTRODUCTION – HISTORICAL REVIEW………………………………1 2) STIRLING CYCLE………………………………………………………….2 3) IDEAL ANALYSIS OF MINIATURE STIRLING CRYOCOOLER………4 4) SCHMIDT CYCLE…………………………………………………………..7 5) SIMULATION WITH SCHMIDT CYCLE………………………………….9 6) RESULTS AND DISCUSSIONS…………………………………………..10 7) SHORTCOMINGS OF SCHMIDT ANALYSIS…………………………..20 8) SCOPE FOR FUTURE WORK…………………………………………….21 9) REGENERATOR OPTIMIZATION……………………………………..22 10) OPTIMIZATION ANALYSIS……………………………………………23 11) OPTIMIZATION PROCEDURE………………………………………..29 12) GRAPHS …………………………………………………………………34 13) CONCLUSION……………………………………………………………42 14) REFERENCES……………………………………………………………43 15) APPENDIX…………………………………………………………….…44 Page | v List of Figures 1) Figure 1: Ideal Stirling Cycle 2) Figure 2: Pressure vs Total Volume 3) Figure 3: Pressure vs Expansion and Compression Volume 4) Figure 4: Pmax vs Tc 5) Figure 5: Qe vs TC 6) Figure 6: W vs TC 7) Figure 7: COP vs TC 8) Figure 8: Pmax vs alpha 9) Figure 9: W vs alpha 10) Figure 10: COP vs alpha 11) Figure 11: Qe vs alpha 12) Figure 12: Pmin vs y (VD/VE) 13) Figure 13: Qe vs y (VD/VE) 14) Figure 14: W vs y (VD/VE) 15) Figure 15:COP vs y (VD/VE) 16) Figure 16: Pmax vs x (VC/VE) 17) Figure 17: COP vs x (VC/VE) 18) Figure 18: W vs x (VC/VE) 19) Figure 19: Qe vs x (VC/VE) 20) Figure 20: Regenerator Porosity vs Qc (Watts) 21) Figure 21: Regenerator Porosity vs Qreg (Watts) 22) Figure 22: Regenerator Porosity vs Wpf (Watts) 23) Figure 23: Regenerator Porosity vs Wprv (Watts) 24) Figure 24: Frequency vs Qc 25) Figure 25: Frequency vs Qreg 26) Figure 26: Frequency vs Qpf 27) Figure 27: Frequency vs Qprv 28) Figure 28: Tc vs Qc 29) Figure 29: Tcold vs Qreg 30) Figure 30: Tcold vs Qpf 31) Figure 31: Tcold vs Qprv 32) Figure 32: Volume Ratio (Vc/Ve) vs Qc 33) Figure 33: Volume Ratio (Vc/Ve) vs Qreg 34) Figure 34: Volume Ratio (Vc/Ve) vs Qpf 35) Figure 35: Volume Ratio (Vc/Ve) vs Qprv Page | vi List of Tables: 1) Table 1: : Stirling Cycle Refrigerator Design Parameters 2) Table 2: First Estimate Of Losses in a Stirling Cryocooler 3) Table 3: Calculated Regenerator Parameters from the Optimization Equations 4) Table 4: Calculated Regenerator Heat Transfer Parameters from Optimized Regenerator Values 5) Table 5: Calculated Regenerator Losses Page | vii NOMENCLATURE: 1) Aff = Fluid axial free flow area 2) Am = Matrix thermal conduction heat transfer area 3) Ar = Total regenerator frontal area 4) As = Matrix total heat transfer area 5) Cc = Cold fluid heat capacity rate 6) Ch = Warm fluid heat capacity rate 7) Cm = Matrix heat capacity rate 8) Cmax = The larger of Cc and Ch 9) Cmin = The smaller of Cc and Ch\ 10) Cp = Specific heat at constant pressure 11) Cv = Specific heat at constant volume 12) Dd = Displacer diameter 13) Dh = Hydraulic diameter (Dh = 4*rh) 14) Dr = Regenerator diameter 15) d = Screen wire diameter 16) f = Fluid coefficient of friction 17) fr = Frequency 18) G = Mass flow rate per unit free flow area 19) h = Heat transfer coefficient 20) Ie = Regenerator thermal inefficiency 21) Kf = Fluid thermal conductivity 22) Km = Matrix thermal conductivity 23) L = Regenerator length 24) M = Molecular weight 25) Mf = Mass of the fluid 26) Mm = Mass of the matrix material 27) m, = Mass flow rate 28) Pmax = Maximum cycle pressure 29) Pmin = Minimum cycle pressure Page | viii 30) Ps = System pressure 31) Pa = Pressure ratio 32) pso = System pressure amplitude 33) Qc = Heat conduction 34) Qh = Heat transferred between fluid and matrix material 35) Qnet = Net cryocooler refrigeration 36) Qreg = Regenerator thermal losses 37) Ta = Ambient room temperature 38) Tcold = Cold end temperature 39) Tw = Warm end temperature 40) t = Time 41) Ve = Expansion space volume 42) Vcs = Compression space volume 43) Vm = Volume occupies by the matrix material 44) Vr = Regenerator total volume 45) Vrv = Regenerator void volume 46) veo = Expansion space volume amplitude 47) Wpv = Gross mechanical refrigeration produced 48) W = Mechanical power 49) Xd = Displacer position 50) Xp = Compressor piston position 51) S = Expander stroke 52) Pmean = Mean cycle pressure 53) p = instantaneous cycle pressure 54) P max = maximum cycle-pressure 55) P min = Minimum cycle-pressure 56) Pmean = Mean cycle-pressure 57) W = Work Done by compressor 58) Qe = heat transferred to the working fluid in expansion space 59) Qc = heat transferred in the compression space 60) COP = coefficient of performance of the cryocooler Page | ix 61) R = characteristic gas constant of the working fluid 62) TC = Temperature of the working fluid in the compression space 63) TD = Temperature of the working fluid in the dead space 64) TE = Temperature of the working fluid in expansion space 65) VC = swept volume of the compression space 66) VE = swept volume of the expansion space 67) VD = swept volume of the dead space 68) x = VC/VE, swept volume ratio 69) y = VD/VE, dead volume ratio 70) t = TC/TE, temperature ratio 71) VT = (VE + VC + VD) 72) α = angle by which volume variations in expansion space lead those in the compression space 73) A = a factor [t2 +2*t*x*cos(α)+x2]0.5 74) S = (VD*TC)/(VE*TD) 75) B = a factor (t+x+2*S) 76) δ =A/B = [ t2 + x2 + 2*t*x*cos(α) ]0.5/(t+x+2*S) 77) θ = atan[{x*sin(α)}]/(t+k+2*S) 78) φ= crank angle Page | x INTRODUCTION HISTORICAL REVIEW A Stirling engine operates in a closed thermodynamic regenerative cycle with the same working fluid repeatedly compressed and expanded at different temperature levels so there is a net conversion of heat to work or vice versa. It can be used as a cooling engine, prime mover, heat pump, or pressure generator. As a cooling machine, it extends back as far as 1817. In 1834 John Herschel conceived the closed cycle regenerative cooling engine for making ice. The concept was not reduced for practice for 30 years. The first Stirling cooling engines were made by Alexander Kirk, a Scottish Engineer working at oil works in Scotland. He constructed cooling engines for a variety of application both in Great Britain and oversees. However the Kirk machines were never made in large numbers or used to achieve cryogenic temperature. The next significant development occurred with the start of the Phillips cooling engine research about 1946. It followed a decade of effort already invested in small Stirling engine prime movers. The Phillips was the first time that substantial effort was invested simultaneously in Stirling engines for power and cooling application.
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